International Archives of Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol. XXXVIII, Part 5
Commission V Symposium, Newcastle upon Tyne, UK. 2010
ERROR PROPAGATION FOR CLOSE-RANGE SINGLE IMAGE RELATIVE
MEASUREMENTS
H. J. Theiss
National Geospatial-Intelligence Agency (NGA), 12310 Sunrise Valley Dr, Reston, VA, USA henry.j.theiss.ctr@nga.mil
Commission V, WG V/1
KEY WORDS: Close range photogrammetry, Resection, Error propagation
ABSTRACT:
NGA uses the Compass software to perform relative measurements on images, usually taken at relatively close range from hand-held
cameras. Compass currently does not have the capability to calculate a confidence interval around its computed relative
measurements. This paper provides some results with both simulated and real data to illustrate how the matrix algebra associated
with a photogrammetric resection can be used to also perform the rigorous error propagation required to derive a confidence interval
around a distance measurement made on a single image. The approach is verified by comparing the actual errors in distance
measurements to the confidence intervals around the estimates.
space rectangular (LSR) object coordinate system. Moreover,
we need to specify the origin of the LSR system, preferably at
an easily identifiable point in the image where three mutually
orthogonal lines intersect. Finally, we need to know at least one
distance in the object space in order for our derived camera
position and any new distance measurements to be in known
ground units, such as meters.
1. INTRODUCTION
The NGA’s Image Chain Analysis (ICA) program uses
collections of handheld imagery to derive three dimensional (3D) Computer Aided Design (CAD) models of features such as
aircraft. Currently, the ICA process does not incorporate error
analysis. Therefore, the imagery derived CAD models have no
means associated with it to aid in determination of the predicted
error and reliability of their measurements. Beginning in
Section 2, this paper describes a photogrammetric approach to
incorporate error propagation into the ICA process. Section 3
then describes experimental results using simulated and real
data. Finally, the paper ends with ideas for future work.
So, the SGC formulation requires a set of condition equations
and constraint equations. The condition equations consist of
the two collinearity equations per observed image point. These
condition equations contain the six EO camera parameters
(common for all points) and the three ground coordinates (X, Y,
and Z for the specific point) as unknowns. The constraint
equations consist of three types. The first type constrains the X,
Y, and Z coordinates of the origin point to be 0, 0, and 0. The
second type constrains each object line to be parallel to the X,
Y, or Z axis. Such constraint requires two equations. For
example, if the line is parallel to the Z axis (a vertical line), then
the difference in X coordinates between the beginning and end
of the line must be 0; and the difference in Y coordinates
between the beginning and end of the line must be 0. The third
type of constraint is a single equation to constrain the distance
between two points to be equal to some measured value.
2. OVERVIEW OF APPROACH
The NGA’s Sensor Geopositioning Centre (SGC) approach to
performing error propagation for a single image is to
mathematically model a resection. A resection is a procedure in
photogrammetry that traditionally assumes that the image
coordinates and associated ground coordinates of several points
are measured, and adjustments to the camera position (XL, YL,
ZL) and attitude (omega, phi, and kappa) are estimated in a least
squares adjustment. These six elements are called the exterior
orientation (EO) parameters. For this report, the camera’s
interior orientation (IO) is known from camera calibration;
however, IO parameters such as focal length and principal point
offset can be included in the resection if the geometry is
sufficiently strong.
In order to calculate the confidence interval for a distance
measurement on an image, we need to first compute the
covariance matrix for the unknown ground coordinates that are
estimated in the resection. Such covariance matrix is the
inverse of the normal equations matrix; see [Mikhail et. al. and
Mikhail] for details. Once the total ground covariance matrix
has been computed, the parts corresponding to the endpoints of
the “feature to be measured” are extracted. The covariance
matrix for the relative vector between point 1 and point 2 is
calculated as follows:
The typical NGA problem involves a close range image with no
known absolute ground coordinates. However, it is often the
case that several orthogonal linear features can be readily
measured in the image. When appropriate, we constrain these
linear features to be parallel to the X, Y, or Z axis of a local
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International Archives of Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol. XXXVIII, Part 5
Commission V Symposium, Newcastle upon Tyne, UK. 2010
 rel  11   22  12  
T
12
Where
deviate from being parallel in image space. An angular field-ofview of 60 degrees (horizontal direction of image format) was
(1) As the field-of-view narrows, these
used in the simulation.
object parallel lines would become more parallel in image
space.
(1)
ij is the 3 by 3 error covariance matrix for the object
space vector between points i and j.
 rel is the 3 by 3 error covariance matrix for the relative
vector between points 1 and 2. If, for example, we know that
such vector is parallel to the Z axis, then we are only interested
in the (3,3) element of the matrix. The one sigma value for the
uncertainty of the distance measurement would then be the
square root of the variance; and the variance is the (3,3) element
of the relative vector error covariance,
 rel .
3. EXPERIMENTAL RESULTS
This section consists of two subsections, one for simulated data
and then subsequently real data.
3.1 Simulated Data
Figure 1 shows the linear features comprising the simulated data
set. Note that some lines were intentionally made shorter than
the others, so that the impact of length of known feature could
be studied.
Figure 2: Image space view of points and lines comprising
simulated data set.
Figure 3 shows the results from the simulated data, particularly
how the measured errors compare to the confidence intervals.
Note the high correlation between the measured and predicted
errors. Also note how infrequently the errors exceed the 90%
confidence interval. The predicted and measured errors become
large for the cases when the relatively short vector is treated as
the known distance.
Figure 1: 3D plot of ground coordinates and lines comprising
simulated data set.
Figure 2 shows the projection of the simulated linear features
into the image space. Note that, due to perspective effects,
parallel lines in object space project to lines that significantly
Figure 3: Measured versus predicted errors for simulated data.
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International Archives of Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol. XXXVIII, Part 5
Commission V Symposium, Newcastle upon Tyne, UK. 2010
Table 1 shows the results from the simulated data compared to
truth, in terms of the mean, standard deviation, RMS, and
confidence intervals (one-sigma and LE90). The results closely
match the numbers expected by statistics; e.g. 68 percent should
fall within one sigma.
3.3 Real Data, Case 2
In this new case, we imposed additional object space constraints
to enforce that certain pairs of lines fell in the same vertical
plane, thus resulting in 19 additional constraint equations.
Also, the focal length was treated as an unknown in this
experiment.
Table 1: Simulated Data Error and Confidence Interval Results
Mean
Standard RMS
% below % below
difference deviation (meters) 1-sigma
LE90
(meters)
(meters)
confidence confidence
interval
interval
0.008
0.028
0.029
67
90
Figure 5 shows the results from the real data using these new
constraints, particularly how the measured errors compare to the
confidence intervals. As in Case 1, note the high correlation
between the measured and predicted errors. Also note how
infrequently the errors exceed the 90% confidence interval.
3.2 Real Data, Case 1
Figure 4 shows the results from the real data, particularly how
the measured errors compare to the confidence intervals. Note
again the high correlation between the measured and predicted
errors. Also note how infrequently the errors exceed the 90%
confidence interval.
Figure 5: Measured versus predicted errors for Case 2.
Table 3 shows the results from the real data compared to truth,
in terms of the mean, standard deviation, RMS, and confidence
intervals (one-sigma and LE90). The results are a little
conservative compared to those numbers expected by statistics;
e.g. 68 percent should fall within one sigma and 90 percent
should fall within the LE90 confidence region.
Figure 4: Measured versus predicted errors for Case 1.
Table 2 shows the results from the real data compared to truth,
in terms of the mean, standard deviation, RMS, and confidence
intervals (one-sigma and LE90). The results closely match the
numbers expected by statistics; e.g. 68 percent should fall
within one sigma and 90 percent should fall within the LE90
confidence region.
Mean
difference
(meters)
-0.01
Mean
difference
(meters)
-1.02
Table 2. Real Data (Case 1) Results
Standard
RMS
% below
deviation (meters)
1-sigma
(meters)
confidence
interval
1.31
1.66
69
% below
LE90
confidence
interval
89
Table 3. Real Data (Case 2) Results
Standard
RMS
% below
deviation (meters)
1-sigma
(meters)
confidence
interval
0.21
0.21
86
% below
LE90
confidence
interval
95
Figure 6 shows the measured and predicted errors as a function
of the magnitude of the measured distance. A noticeable
positive correlation exists, albeit not too great.
Although the error propagation appears to be working correctly,
the magnitude of these predicted and measured errors are much
larger than the authors expected them to be; i.e. on the order of
one to two meters. The next real data case then investigates
what additional constraints can be imposed in the math model
in order to increase the precision.
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International Archives of Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol. XXXVIII, Part 5
Commission V Symposium, Newcastle upon Tyne, UK. 2010
References from Books:
Mikhail, E.M., Bethel, J.S., McGlone, J.C., 2001. Introduction
to Modern Photogrammetry. John Wiley and Sons, New York,
NY.
Mikhail, E.M., 1976.
Observations and Least Squares.
University Press of America, New York, NY.
Acknowledgements:
The author acknowledges, with sincere thanks, the contributions
from and open collaboration with scientists in the SGC as well
as scientists and analysts in other divisions of NGA.
Figure 6: Measured and predicted errors versus distance.
Figure 7 shows the measured and predicted errors as a function
of length of the control line used in establishing the camera
model. A noticeable but subtle negative correlation exists.
Figure 7: Measured and predicted errors versus control line
length.
4. CONCLUSIONS AND FUTURE WORK
The SGC successfully developed a prototype that could perform
rigorous error propagation for a single image, and therefore
provide a confidence interval around any distance measurement.
The measured errors are consistent with the predicted errors.
In the future, in addition to the investigation to improving
precision (discussed in Section 3), the SGC plans to work on
some of the following tasks:

Compare error propagation performance between this
paper’s approach and classical close range
formulations that use vanishing lines.

Develop prototype for deriving confidence intervals
around measurements other than distances.

Develop prototype and/or provide recommendations
for how to analyze the results and error propagation
from the multi bundle analysis (MBA).

Investigate impact of modelling lens distortion.
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